What Is the Probability That K People Will Occupy Adjacent Chairs in a Circle?

[EnzoF61]

Homework Statement

If k people are seated in a random manner in a circle containing n chairs (n>k), what is the probability that the people will occupy k adjacent chairs in the circle?

The Attempt at a Solution

1) The total probability of k people seated in a circle of n chairs is [ (n "choose" k) = p1 ]

2) The amount of people occupying k adjacent chairs. We have (n-k) total unoccupied chairs. For a small chunk of the total number, (Kth Person, 1st Person, 2nd Person are all seated next to each other) or (2nd Person, 1st Person, Kth Person seated adjacently). => (n-k+2)=p2 total ways to seat adjacently (considering the two identical orders).

3) Solution = p2*p1 = (n-k+2) / (n "choose" k)

 

[mighty2000]

Thank you for your question. I would like to offer a different approach to solving this problem.

Firstly, let's define some variables:
- n: total number of chairs in the circle
- k: number of people to be seated
- p: probability of k adjacent chairs being occupied

To solve this problem, we can use the concept of permutations and combinations. Permutations are arrangements of objects where order matters, while combinations are arrangements where order does not matter.

In this case, we are interested in combinations, as the order in which the k people are seated does not affect the outcome. Therefore, we can use the combination formula nCr = n! / r!(n-r)! to calculate the total number of ways the k people can be seated in the circle.

Next, we need to consider the number of ways the k people can occupy k adjacent chairs. This can be calculated by fixing one person in a chair and then rearranging the remaining (k-1) people in the remaining (n-1) chairs, which can be done in (n-1)P(k-1) ways.

Finally, we can calculate the probability p by dividing the number of ways the k people can occupy k adjacent chairs by the total number of ways they can be seated in the circle, which gives us the final formula: p = (n-1)P(k-1) / nCr.

I hope this helps to clarify the solution to this problem. If you have any further questions, please do not hesitate to ask. Best of luck with your studies!